Question: Simplify the following expression and state the condition under which the simplification is valid: $y = \dfrac{r^2 - 8r + 15}{r^2 - 10r + 21}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{r^2 - 8r + 15}{r^2 - 10r + 21} = \dfrac{(r - 5)(r - 3)}{(r - 7)(r - 3)} $ Notice that the term $(r - 3)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(r - 3)$ gives: $y = \dfrac{r - 5}{r - 7}$ Since we divided by $(r - 3)$, $r \neq 3$. $y = \dfrac{r - 5}{r - 7}; \space r \neq 3$